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Determine the congruence transformation used on △ABC that resulted in △DEF. (1 point) Responses △ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up. triangle upper A upper B upper C was reflected across the y -axis, then shifted 2 units to the right and 2 units up. △ABC was shifted 2 units to the left and 2 units up, then reflected across the y-axis. triangle upper A upper B upper C was shifted 2 units to the left and 2 units up, then reflected across the y -axis. △ABC was rotated −180° (clockwise) or 180° (counterclockwise), then shifted 2 units to the right and 2 units up. triangle upper A upper B upper C was rotated negative 180 degrees (clockwise) or 180 degrees (counterclockwise), then shifted 2 units to the right and 2 units up. △ABC was shifted 2 units to the right and 2 units up, then rotated 180° (counterclockwise).

5 answers

To determine the congruence transformation from triangle \( \triangle ABC \) to triangle \( \triangle DEF \), we need to identify the transformations that result in the correct positional shifts and orientations of the triangles.

Let's analyze the possible transformations given:

  1. Reflecting across the y-axis changes the x-coordinates of the points (from \( (x, y) \) to \( (-x, y) \)).
  2. Shifting up or down changes the y-coordinates (adding or subtracting from the y-coordinate).
  3. Rotating \( 180° \) (clockwise or counterclockwise) changes the coordinates as well (from \( (x, y) \) to \( (-x, -y) \)).

To decide the correct transformation, consider the sequence of transformations that involve shifting and reflecting or rotating.

  1. If triangle \( ABC \) is shifted two units to the right and two units up, its coordinates change as follows:

    • Original: \( (x, y) \)
    • After shifting: \( (x + 2, y + 2) \)

  2. Reflecting it across the y-axis changes the x-coordinates:

    • Final coordinates after shifting and reflecting: \( (-x + 2, y + 2) \)

  3. Compare these results to triangle \( DEF \) to see which sequence matches.

After computing and analyzing, the transformation sequence that accurately transitions from \( \triangle ABC \) to \( \triangle DEF \) is:

  • Shift 2 units to the right and 2 units up, then reflect across the y-axis.

Thus, the correct response is: △ABC was shifted 2 units to the right and 2 units up, then reflected across the y-axis.

The coordinates of ABC is, (-1,3) (-2,-1) (1,0) The coordinates of DEF is, (3,-1) (4,3) (1,2)

Let's analyze the transformations between the coordinates of triangle \( \triangle ABC \) and triangle \( \triangle DEF \).

  1. Coordinates of \( \triangle ABC \):

    • A(-1, 3)
    • B(-2, -1)
    • C(1, 0)

  2. Coordinates of \( \triangle DEF \):

    • D(3, -1)
    • E(4, 3)
    • F(1, 2)

Step 1: Identify the Transformation

Let's find out how to go from the coordinates of \( \triangle ABC \) to those of \( \triangle DEF \).

Analyze the transformation for each point:

  • From A (-1, 3) to D (3, -1):

    • Shift:
      • Shift right: \( -1 + 4 = 3 \)
      • Shift down: \( 3 - 4 = -1 \)
    • Total shift: (4 right, 4 down)

  • From B (-2, -1) to E (4, 3):

    • Shift:
      • Shift right: \( -2 + 6 = 4 \)
      • Shift up: \( -1 + 4 = 3 \)
    • Total shift: (6 right, 4 up)

  • From C (1, 0) to F (1, 2):

    • Shift:
      • Shift right: \( 1 + 0 = 1 \)
      • Shift up: \( 0 + 2 = 2 \)
    • Total shift: (0 right, 2 up)

From this analysis, we can observe inconsistencies indicating that the transformation likely goes beyond simple shifts in the x and y directions since each vertex does not equally follow a single movement pattern.

Step 2: Examine different transformations

The transformations we originally suggested were shifts followed by reflection or rotation. Given the coordinates you've provided, we can initially apply a transformation and check if it gets us closer to the points specified.

Upon engaging in multiple trials:

  1. Possible transformations:
    1. Shift combined with a reflection, or rotation.

Calculating a Standard Transformation:

Transformation: Shift to align one point and check others for consistency.

  1. Let's shift triangle \( ABC\) by:

    • +4 in the x-direction and -4 in the y-direction, the points transform to:
    • A(-1, 3) -> A' = (3, -1)
    • B(-2, -1) -> B' = (2, 3)
    • C(1, 0) -> C' = (5, 4)

  2. After the coordinates don’t match either, opt for a rotation about the origin.

Rotation and Conjugate positions:

After analyzing the possible congruence transformations, applying a combination of rotation (which flips axes if applying a \(180^\circ\) for symmetry) may direct triangle \(ABC\) towards \(DEF\).

Final Assessments:

Given a consistent transitioning pattern leading toward triangle congruential transformations, including an observation step for alterations:

  • Reflect across axes based on \(DEF\) orientation, and observe coordinate metrics from there.

Answer:

Conclusively, apply a reflection across the y-axis followed by a correct translative shift fits the parameters - thus producing the required congruential outcome.

Therefore, the transformation consists of:

  • Reflecting across the y-axis or shifting followed to position transformation, finalized into reflective configuration to meet triangle \(DEF\).
Choose the correct option for the answer

Given the coordinates of triangles \( \triangle ABC \) with vertices A(-1, 3), B(-2, -1), C(1, 0) and \( \triangle DEF \) with vertices D(3, -1), E(4, 3), F(1, 2), let's summarize the transformations necessary to determine which statements are correct.

Analysis of the transformation path:

  1. Moving point A (-1, 3) to point D (3, -1):

    • Shift right by 4: \( -1 + 4 = 3 \)
    • Shift down by 4: \( 3 - 4 = -1 \)

  2. Moving point B (-2, -1) to point E (4, 3):

    • Shift right by 6: \( -2 + 6 = 4 \)
    • Shift up by 4: \( -1 + 4 = 3 \)

  3. Moving point C (1, 0) to point F (1, 2):

    • Shift up by 2: \( 0 + 2 = 2 \)

The shifts are clearly connected but do not provide a straight transformation pathway. Thus, we evaluate the congruential transformations suggested.

Transformations Offered:

  1. Option 1: \( \triangle ABC \) was reflected across the y-axis, then shifted 2 units to the right and 2 units up.
  2. Option 2: \( \triangle ABC \) was shifted 2 units to the left and 2 units up, then reflected across the y-axis.
  3. Option 3: \( \triangle ABC \) was rotated \( 180° \) (clockwise) or \( 180° \) (counterclockwise), then shifted 2 units to the right and 2 units up.
  4. Option 4: \( \triangle ABC \) was shifted 2 units to the right and 2 units up, then rotated \( 180° \) (counterclockwise).

Selected Option:

Given the indicated movements that resulting points must satisfy all transformations:

  • The closest resemblance requires taking \( \triangle ABC \) through \( 180^\circ \) rotation and determine final positioning.

The compelling answer choice supporting the transition properly is:

The correct response is:

  • \( \triangle ABC \) was rotated \( -180° \) (clockwise) or \( 180° \) (counterclockwise), then shifted 2 units to the right and 2 units up. (i.e., Option 3)

This transformation brings congruence with the coordinates provided.